RIO DE JANEIRO — When Simone Biles, the supernova of the U.S. women’s gymnastics team, stuck the landing on a near-flawless balance beam routine at Rio Olympic Arena, she let out a squeal.

With it, Biles sealed an American rout of Sunday’s qualifying sessions and bounced directly into the embrace of four adoring teammates. U.S. national team director Martha Karolyi told her gymnasts that she loved them. And when informed of their final margin — a whopping 10 points ahead of China, their nearest competitor — the 73-year-old Karolyi jumped up and down and said, “I love you even more!”

The Americans won no medals for Sunday’s display of might. Qualifying serves only to cull the world’s gymnasts for the Olympic medal rounds in store — sending the top eight nations into Tuesday’s team final, the top 24 individual performers on to Thursday’s prestigious all-around final and the top eight gymnasts on each apparatus on to later finals on floor exercise, vault, uneven bars and beam.

With the 4-foot-9 Biles leading the way — scoring event-high marks on three of the sport’s four apparatus (floor, vault and beam) — the U.S. women emerged as the most prohibitive Olympic gold-medal favorite since USA Basketball’s Dream Team of 1992.

The U.S. women routed all to finish with 185.238 points to China’s 175.279, followed by Russia’s 174.620.

Meet the five women representing the United States in the 2016 Olympic games. (Ashleigh Joplin,Jayne Orenstein/The Washington Post)

In terms of individual marks, the Americans earned the top three score, which would have meant a gold-silver-bronze haul, had qualifying counted for medals and the all-around competition a true meritocracy. But under Olympic rules, it meant disappointment for reigning Olympic all-around champion Gabby Douglas.

That’s because each nation may qualify no more than two gymnasts for the individual all-around. The same is true for the event finals.

So though Douglas posted the third-highest score among the 2016 Olympians, she missed the cut to defend her 2012 all-around gold because two teammates — Biles and Aly Raisman, 22, the team’s captain — finished ahead of her .

Biles’s final score was 62.366. Raisman scored 60.601, and Douglas was 0.470 behind, with 60.131.

“I really would have loved to go out there and defend my title; however, you know that it is what it is,” Douglas said. “I’m still rejoicing. I’m not ashamed at all. I’m pretty pleased with everything overall, and I’s still rejoicing because there’s nothing to be ashamed about.”

U.S. Olympic Coach Aimee Boorman lamented the two-per-country limit.

“I don’t like it because I feel like if it’s the best athletes, the best athletes should be competing,” said Boorman, Biles’s longtime coach. “I would even feel that way if our kids were not in the top eight.”

While Biles posted the highest score on three events, fellow American Madison Kocian earned the top score on the uneven bars, her signature event. All five U.S. women qualified for at least one apparatus final.

Ranging in age from 16 to 22, the gymnasts celebrated their display of dominance in a manner rarely seen on fields of play. There was no fist-pumping nor shrieks.

“I think today we made everyone proud,” said Biles, describing herself as “super-excited.” “And we all qualified for an event, so I don’t think Martha could have asked for anything better.”

Olympic qualifying doesn’t call for gymnasts to give the performance of their lives; it’s far better to deliver error-free routines that ensure their country a spot in the team finals and, ideally, lock up a spot in individual events.

With gymnasts from multiple nations competing on all four events at once, there was something to see in every quadrant of the arena floor Sunday. But the Americans simply demanded to be watched.

Biles, the 19-year-old Texan who has raised the standard of her sport with the daring, precision and utter joyfulness of her skills, brought the crowd to its feet with the sky-high tumbling of her Brazilian-flavored floor routine.

She followed with two lights-out vaults: a high flying, technically tricky Amanar, followed by an even more difficult Cheng, which she landed like a still-life painting — frozen in space, without a wobble or waver.

Douglas wasn’t at her best on floor or vault but proved her value to the team on uneven bars, whipping around them with elegance. Kocian, a reigning world co-champion on uneven bars, closed the rotation with an even more daring routine, executed with grace and precision.

Entering the balance beam, the Americans’ final event, Raisman held a 0.476-point lead on Douglas, which effectively meant that their performance on beam would determine who would join Biles in the all-around final.

Douglas had a slight balance-check but stayed on display impressive tumbling skills. Raisman had a more pronounced bobble, leaning to one side before righting herself, but gave judges nothing to quibble with otherwise. The two gymnasts ended up with identical beam scores, which clinched the all-around spot for Raisman.

While Raisman was overjoyed to qualify for another chance at an all-around medal, having been edged for bronze in 2012, she called the two-per-country rule “heart-breaking,” saying she believed all five members of the U.S. team were among the world’s top 24 gymnasts.

“I wish they would change it, but I don’t think they ever will,” Raisman said.